G
Topic
First Poster
Last Poster
Weird expression being integer.
MarkBcc168 24
N
by AR17296174
Source: IMO Shortlist 2017 N5
Find all pairs 
of prime numbers which 
and
is an integer.
of prime numbers which 
and
is an integer.
24 replies
Find the value of n - ILL 1990 MEX1
Amir Hossein 3
N
by maromex
During the class interval, 
children sit in a circle and play the game described below. The teacher goes around the children clockwisely and hands out candies to them according to the following regulations: Select a child, give him a candy; and give the child next to the first child a candy too; then skip over one child and give next child a candy; then skip over two children; give the next child a candy; then skip over three children; give the next child a candy;...
Find the value of for which the teacher can ensure that every child get at least one candy eventually (maybe after many circles).
children sit in a circle and play the game described below. The teacher goes around the children clockwisely and hands out candies to them according to the following regulations: Select a child, give him a candy; and give the child next to the first child a candy too; then skip over one child and give next child a candy; then skip over two children; give the next child a candy; then skip over three children; give the next child a candy;...Find the value of
for which the teacher can ensure that every child get at least one candy eventually (maybe after many circles).
3 replies
Self-evident inequality trick
Lukaluce 14
N
by sqing
Source: 2025 Junior Macedonian Mathematical Olympiad P4
Let 
, and 
be positive real numbers, such that 
. Prove the inequality
![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](//latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
, and 
be positive real numbers, such that 
. Prove the inequality![\[\frac{x^3}{2 + x} + \frac{y^3}{2 + y} + \frac{z^3}{2 + z} \ge 1.\]](http://latex.artofproblemsolving.com/b/e/5/be5819a67c3cd78f2dea35fdccf48688c720ce3c.png)
When does the equality hold?
14 replies
+1 w
maximum value
moldovan 4
N
by MathIQ.
Source: Austria 1989
Natural numbers 
satisfy 
. Find the maximum value of the product 
satisfy 
. Find the maximum value of the product 
4 replies
R to R, with x+f(xy)=f(1+f(y))x
NicoN9 5
N
by jasperE3
Source: Own.
Find all functions 
such that![\[
x+f(xy)=f(1+f(y))x
\]](//latex.artofproblemsolving.com/4/3/3/433f95836dad1224754d480aff8432ef033a37cb.png)
for all 
.
such that![\[
x+f(xy)=f(1+f(y))x
\]](http://latex.artofproblemsolving.com/4/3/3/433f95836dad1224754d480aff8432ef033a37cb.png)
for all 
.
5 replies
Chain of floors
Assassino9931 1
N
by pi_quadrat_sechstel
Source: Vojtech Jarnik IMC 2025, Category I, P2
Determine all real numbers 
such that
![\[ \left\lfloor\frac{n+1}{x}\right\rfloor = n - \left\lfloor \frac{n}{x} \right\rfloor + \left \lfloor \frac{\left \lfloor \frac{n}{x} \right\rfloor}{x}\right \rfloor - \left \lfloor \frac{\left \lfloor \frac{\left\lfloor \frac{n}{x} \right\rfloor}{x} \right\rfloor}{x}\right \rfloor + \cdots \]](//latex.artofproblemsolving.com/3/1/9/319c9a2a80c4da5961cd5389af0606fa429ca8e2.png)
for any positive integer .
such that![\[ \left\lfloor\frac{n+1}{x}\right\rfloor = n - \left\lfloor \frac{n}{x} \right\rfloor + \left \lfloor \frac{\left \lfloor \frac{n}{x} \right\rfloor}{x}\right \rfloor - \left \lfloor \frac{\left \lfloor \frac{\left\lfloor \frac{n}{x} \right\rfloor}{x} \right\rfloor}{x}\right \rfloor + \cdots \]](http://latex.artofproblemsolving.com/3/1/9/319c9a2a80c4da5961cd5389af0606fa429ca8e2.png)
for any positive integer .
1 reply
2015 Taiwan TST Round 2 Quiz 1 Problem 2
wanwan4343 8
N
by Want-to-study-in-NTU-MATH
Source: 2015 Taiwan TST Round 2 Quiz 1 Problem 2
Let 
be the incircle of triangle 
and touches 
at 
. 
meets again at 
. Let 
be 
-excenter, and 
be the midpoint of 
, respectively. Prove that 
are concyclic.
be the incircle of triangle 
and touches 
at 
. 
meets again at 
. Let 
be 
-excenter, and 
be the midpoint of 
, respectively. Prove that 
are concyclic.
8 replies
Prove that
Triangle form by perpendicular bisector
psi241 52
N
by ihatemath123
Source: IMO Shortlist 2018 G5
Let be a triangle with circumcircle 
and incentre 
. A line 
intersects the lines 
, 
, and 
at points , 
, and 
, respectively, distinct from the points , 
, 
, and . The perpendicular bisectors 
, 
, and of the segments , 
, and 
, respectively determine a triangle 
. Show that the circumcircle of the triangle is tangent to .
be a triangle with circumcircle 
and incentre 
. A line 
intersects the lines 
, 
, and 
at points , 
, and 
, respectively, distinct from the points , 
, 
, and . The perpendicular bisectors 
, 
, and of the segments , 
, and 
, respectively determine a triangle 
. Show that the circumcircle of the triangle is tangent to .
52 replies
Regional Olympiad - FBH 2018 Grade 9 Problem 3
gobathegreat 5
N
by justaguy_69
Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2018
Let 
and 
be prime numbers such that 
is perfect square. Prove that 
is prime
and 
be prime numbers such that 
is perfect square. Prove that 
is prime
5 replies
